LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering

TL;DR

This work tackles graph clustering without a predefined number of clusters by introducing differentiable structural information (DSI) and a Lorentz-space neural network, LSEnet. By relaxing discrete structural-entropy concepts into a differentiable, level-wise formulation, the method learns a partitioning tree that reveals cluster structure while minimizing conductance, without requiring a fixed $K$. LSEnet embeds nodes in hyperbolic space via a Lorentz-convolutional architecture and builds the partitioning tree bottom-up with a Lorentz assigner, integrating node features through hyperbolic graph convolutions. Empirical results on standard graphs show strong clustering performance, favorable scalability compared with classic structural entropy, and interpretable hyperbolic trees visualized on real data. The approach offers a scalable, geometry-aware pathway for clustering graphs with unknown cluster counts while preserving rich feature information.

Abstract

Graph clustering is a fundamental problem in machine learning. Deep learning methods achieve the state-of-the-art results in recent years, but they still cannot work without predefined cluster numbers. Such limitation motivates us to pose a more challenging problem of graph clustering with unknown cluster number. We propose to address this problem from a fresh perspective of graph information theory (i.e., structural information). In the literature, structural information has not yet been introduced to deep clustering, and its classic definition falls short of discrete formulation and modeling node features. In this work, we first formulate a differentiable structural information (DSI) in the continuous realm, accompanied by several theoretical results. By minimizing DSI, we construct the optimal partitioning tree where densely connected nodes in the graph tend to have the same assignment, revealing the cluster structure. DSI is also theoretically presented as a new graph clustering objective, not requiring the predefined cluster number. Furthermore, we design a neural LSEnet in the Lorentz model of hyperbolic space, where we integrate node features to structural information via manifold-valued graph convolution. Extensive empirical results on real graphs show the superiority of our approach.

Figures (6)

• Figure 1: Overview. In hyperbolic space, the proposed LSEnet learns a partitioning tree for node clustering without predefined $K$.
• Figure 2: The overall architecture of LSEnet. We encode $G$ into Lorentz model $\mathbb{L}^{\kappa, d_{\mathcal{T}}}$ via a Lorentz convolution layer, and recursively utilize Lorentz assignor and geometric centroid to construct a tree from down to up. Optimizing DSI, we learn the optimal $\mathcal{T}_{\operatorname{net}}^*$ in hyperbolic space, and obtain clustering results correspondingly.
• Figure 3: Parameter sensitivity on dimension of structural entropy.
• Figure 4: Running time to obtain partitioning tree
• Figure 5: Visualization of hyperbolic partitioning trees.

Theorems & Definitions (27)

• Definition 4.1: $H$-dimensional Structural Entropy li2016structural
• Definition 4.2: Level-wise Assignment
• Definition 4.3: H-dimensional Structural Information
• Theorem 4.4: Equivalence
• proof
• Lemma 4.5: Additivity-Appendix \ref{['proof.equivalence']}
• Theorem 4.6: Connection to Graph Clustering-Appendix \ref{['proof.conductance']}
• proof
• Theorem 4.7: Bound-Appendix \ref{['proof.theoremapprox']}
• Theorem 4.8: Flexiblity-Appendix \ref{['proof.flex']}